Musical Applications of Complex Geometries: Hexany, Orbifold Spaces, and Voronoi Diagrams

Ryan McGee | MAT 594CM | Spring 2009

HEXANY

Hexany is an example of musical geometry in which a six-note just intonation scale is derived from notes placed on the vertices of an octahedron. Every edge of the octahedron joins notes that make a consonant dyad, and every face joins notes that form a consonant triad.

Tuning is usually accomplished by choosing four prime numbers and extracting their six musically unique combinations- one for each vertex of the octahedron.

Click the above image for an interactive example of hexany by Robert Walker

It is also possible to extend the concept of hexany beyond 3D space (using hypercubes) to create more complex chord structures.

Detailed Hexany Notes (found on Wikipedia, unknown author)

MAPPING MUSIC TO ORBIFOLD SPACES (Dmitri Tymoczko)

Dmitri Tymoczko has pioneered a method of mapping music to orbifold spaces. The orbifolds he uses have anywhere from 2 to infinite dimensions depending on the number of musical notes being played at once. Chords that are pleasing to ear (consonant) appear close to each other towards the center of the orbifold space, while displeasing (dissonant) sounds appear near the edges of the orbifold. Tymoczko reminds us that playing the piano is a way of interacting with non-euclidean space since there are several ways to play any given chord. Thus, it becomes somewhat intuitive to map music to a non-euclidean orbifold rather than the traditional restricting spaces such as the staff or cicrcle of fifths. His technique has been implemented in software used for analysis of pieces as well as a compositional tool.

Simply put, when mapping chords to an orbifold space consider that an ordered sequence of n pitches can be mapped as a point in Rn. Line segments between points in this space represent possible voice leadings and the measurements of these line segments can be used to determine perceptual chord similarity. Tymoczko defines three types of chord symmetry within orbifold spaces:

· T-symmetry (transpositional) – a chord that divides the octave into equal parts or is the union of equally sized subsets that do so. There is increasing t-symmetry as one moves near the center of the orbifold. Composers tend to prefer t-symmetry.

· P-symmetry (permutattionally) – a chord with a duplicate pitch class. Chords with p-symmetry lie on the singular boundaries of the orbifold. Perfect p-symmetry would result in a completely dissonant and unpleasing sound. However, high amounts of near p-symmetry are found in the works of avant-garde composers such as Ligeti who work with lots of dissonant sounds.

· I-symmetry (inversionally) – a chord that is invariant under reflection in pitch class space. These are found throughout the orbifold space.

Distances between chords with the above symmetries relate to perceptual chord similarity. Thus, most compositions contain chords lying the same region of orbifold space.

Tymoczko’s paper on the Geometry of Musical Chords

MUSICAL APPLICATIONS OF VORONOI DIAGRAMS (Alex McLean)

1) A visual reference for live music:

Voronoi diagrams can be computed to accompany live music events with relatively low latency using a Fortune sweep-line. While Voronoi vertices can not be calculated in real-time along the seep line, the Fortune method allows one to calculate vertices with certainty halfway between the sweep line and closest site Using one axis of a 2D voronoi diagram as time one can place the sweep line on this axis and progressively calculate the voronoi diagram in near real time.

Example of Voronoi diagram computed for live music. Notice the sweep line at the right side.

When a musical event occurs, it is represented as a Voronoi site at the sweep-line, and the site’s position on the Y axis is chosen according to the

instrument and pitch. The regular events shown at the bottom of the diagram represent a simple

rhythm, and the clustering patterns above represent a melodic structure.

2) Automatic Application of Effects to Groups of Musical Events:

From Alex Norman’s paper Voronoi Diagrams of Music –

“As we have seen, Voronoi diagrams allow us to identify clusters of musical events, and calculate

metrics revealing some data about the topology of such clusters. This could allow effects that react

to the structure of the musical events behind the sounds that are being effected. For example,

a dense cluster of notes might be enhanced so that each sound can be better distinguished, and

more widely spaced sounds might have some time varied modulation applied to accentuate their

presence. This process of effecting sounds based on Voronoi metrics would be akin to sonification

of the diagram, the result being an accentuation of topological features of the music.”

3) Spacing of Musical Events

Using centroidal Voronoi tesselations (CVT), a Voronoi tessellation in which the generating point of each cell is also the center of mass, McLean quantized a simple melody according to several iterations of CVT. Visual and audio results of the experiment are below.

Original melody original.mp3

Melody after one iteration step_one.mp3

Melody after two iterations step_two.mp3

Melody after three iterations step_three.mp3

Melody after one hundred iterations step_hundred.mp3

Alex McLean’s Paper on Voronoi Diagrams of Music